On lattices embeddable into subsemigroup lattices. III: Nilpotent semigroups

We prove that the class of the lattices embeddable into subsemigroup lattices of $n$-nilpotent semigroups is a finitely based variety for all $n<\omega$. Repnitskii showed that each lattice embeds into the subsemigroup lattice of a commutative nilsemigroup of index 2. In this proof he used a result of Bredikhin and Schein which states that each lattice embeds into the suborder lattices of an appropriate order. We give a direct proof of the Repnitskii result not appealing to the Bredikhin–Schein theorem, so answering a question in a book by Shevrin and Ovsyannikov.